Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36468 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,40,37);(1,2)(3,6)(4,7)(5,8)(9,10,14,12)(11,15,18,17)(13,20,21,16)(19,23,24,22)(25,27)(26,31)(28,37)(29,39)(30,36)(32,40)(33,38)(34,35);(1,3,5,4)(2,6,8,7)(10,12)(11,18)(16,20)(19,24)(27,36)(28,38)(31,35)(32,39);(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,17,18,15)(13,16,21,20)(19,22,24,23)(25,28,30,38)(26,32,34,39)(27,37,36,33)(29,31,40,35);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,35,38,31)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7879021);